The superior properties of the laser as a light source have revolutionized optics in a wide variety of applications ranging from science and medicine on the one hand to optical communications and CD players on the other. A laser includes two basic components: an active region and an optical resonator. When suitably pumped by an energy source, the active region generates light at a center wavelength determined by the active region material or its structure. The resonator, which contains the active region and provides optical feedback for the stimulated emission of light, influences the special characteristics of the emitted light; e.g., its optical power, beam directionality and spectral properties. The resonator also determines the physical features of the laser such as its size and shape.
Semiconductor lasers, the most widely used and versatile class of lasers, typically employ resonator mirrors in the form of either cleaved crystal facets (Fabry-Perot cavities), etched distributed feedback (DFB) gratings, etched distributed Bragg reflectors (DBRs), or a suitable combination of them. In general, it is desirable to increase the reflectivity of the resonator mirrors in order to reduce the lasing threshold and the volume of the active region. Satisfying these desiderata reduces the pump energy required and increases the packing density capability of the lasers (e.g., in an array or an optical IC).
The prior art has made significant advances in the development of high reflectivity mirrors, particularly in the relatively recent, innovative design of microdisk lasers. These lasers, which exploit total internal reflection (TIR) of light to achieve mirror reflectivity very close to unity, are based on circularly symmetric (e.g., cylindrical or disk-like) resonators. These resonators support lasing in what is known as whispering gallery (WG) modes. See, for example, S. L. McCall et al., Appl. Phys. Lett., Vol. 60, No. 3, pp. 289-291 (1992), which is incorporated herein by reference. In a WG mode, as shown in FIGS. 3A (right inset) and 5A, light circulates along a modal path that stays near to the curved cylindrical boundary of the resonator, reflecting from the walls of the resonator at an angle of incidence always larger than the critical angle for TIR. Thus, essentially all of the circulating light remains trapped inside the resonator, with only minute losses of light due to evanescent leakage through the boundary (i.e., tunneling) and due to scattering from roughness on the wall surfaces. Significant additional characteristics of WG modes relate to their angle of incidence at the resonator boundary and their sense of rotation within the resonator. More specifically, the angle of incidence, .chi., is conserved. That is, the WG mode always impinges on the boundary at the same angle such that sin .chi..gtoreq.1/n. And, the sense of rotation for a light ray propagating along a particular modal path is constant in time and fixed in space; e.g., it is either clockwise or counter clockwise along a given modal path, and it does not change its sense of rotation with time. See, for example, J. U. Noeckel et al., Optical Processes in Microcavities, R. K. Chang et al., Eds. (World Scientific Publishers, Singapore, 1995), Ch. 11 entitled Chaotic Light: A Theory of Asymmetric Resonant Cavities (hereinafter, Noeckel 95), which is incorporated herein by reference.
Serious disadvantages of microdisk lasers based on WG modes, however, include relatively low output power (in the range of a microwatt for mid-infrared quantum cascade microdisk lasers) due to the high Q of the resonator, and the lack of directional output emission due to the circular symmetry. Thus, a need remains in the art for a microdisk laser design that provides relatively high output power as well as output beam directionality.
Recent theoretical work on WG mode resonators formed in relatively low refractive index materials (n&lt;2) has addressed the issue of directional emission. See, for example, J. U. Noeckel et al., Nature, Vol. 385, No. 6611, pp. 45-47 (1997; hereinafter Noeckel 97), J. U. Noeckel et al., Opt. Lett., Vol. 21, No. 19, pp. 1609-1611 (1996, hereinafter Noeckel 96), J. U. Noeckel et al., Opt. Lett., Vol. 19, No. 21, pp. 1693-1695 (1994, hereinafter Noeckel 94), all of which are incorporated herein by reference, and Noeckel 95, supra. The resonators studied were asymmetric resonant cavities (ARCs), which are WG resonators with weak deformations from circular cylindrical (or spherical) symmetry. The ray dynamics of these deformed resonators is either partially or fully chaotic in the generic case. See, Noeckel 95, supra. In a chaotic resonator, for a large fraction of the ray trajectories (i.e., the orbits or modal paths corresponding to given modes), the trajectory of a subsequent ray, which differs in launch conditions (i.e., starting point and launch angle) by even the smallest amount from an original ray, cannot be predicted from the launch conditions of the original ray.
The type of deformation studied in greatest depth in this body of theoretical work is a two-dimensional convex resonator with a quadrupolar deformation of the circular boundary, described in polar coordinates (r,.phi.) by the following expression: EQU r(.phi.){character pullout}(1+.epsilon. cos 2.phi.) (1)
where .epsilon. is the deformation parameter. Equation (1) implicitly defines a coordinate system where .phi.=0.degree. corresponds to the direction of highest curvature and is oriented parallel to the major (elongated) axis of the deformed cross-section. Partially chaotic WG modes in these resonators have shown directional lasing emission in relatively low refractive index materials (n&lt;2; e.g., glass fibers or cylindrical dye jets). See, Noeckel 96, supra. The origin of the directional emission is found in Noeckel 97, supra. That is, the deformed boundary causes the angle of incidence of a ray in a WG mode to fluctuate in time and on average to increase. Eventually a ray initially trapped by TIR impinges on the boundary below the critical angle and escapes by refraction. The direction of emitted light can in principle be controlled by a suitable choice of deformation parameter. But, this work did not consider the effects associated with higher index materials (e.g., n&gt;3.3 typical of Group III-V compound semiconductor lasers). In particular, it did not demonstrate how to obtain higher output power in semiconductor microdisk lasers.